Vector space

Submitted by Isoscel on Fri, 11/30/2007 - 19:44.

Here is the definition of a vector space over a field $\mathcal{K}$ .
Let $(\mathcal{K},+,.)$ be a field and ( $V$ ,+) a commutative group.
$(i)\forall x,y,z \in V, x+(y+z)=(x+y)+z$
$(ii)\exists \:0_V\in V$ such as $\forall x\in V, x+0_V=0_V+x=x$
$(iii)\forall x\in V, \exists -x\in V,$ such as $x+(-x)=(-x)+x=0_V$
$(iv)\forall x,y \in V, x+y=y+x=0_V$
Let * be a binary operation $*:\mathcal{K}\times V\rightarrow V$ such as
$(v)\forall \alpha ,\beta \in \mathcal{K} x\in V, (\alpha +\beta )*x= \alpha *x+\beta* x$
$(vi)\forall \alpha \in \mathcal{K},\: x,y \in V, \alpha *(x+y)= \alpha *x+\alpha *y$
$(vii)\forall \alpha ,\betha \in \mathcal{K} x\in V, \alpha *(\beta *x)= (\alpha \beta) *x$
$(viii)\forall x\in V, 1_{\mathcal{K}}*x=x$
Then $(V,+,*)$ is called a vector space over field $\mathcal{K}$ or simply a $\mathcal{K}$ -vector space.

Morphism of Vector Space

Let $(V_1,+,*)$ and $(V_2,+,*)$ two $\mathcal{K}$ -vector spaces.
A function $f:V_1\rightarrow V_2$ is a $\mathcal{K}-$ morphism of $\mathcal{K}$ -vector spaces if
$\forall \alpha ,\beta \in \mathcal{K} ,\:x,y \in V,$ ,

$f(\alpha *x+\beta *y)=\alpha *f(x)+\beta *f(y)$

Subspace of a vector space

Let $(V,+,*)$ be a $\mathcal{K}$ -vector space. A subset $\mathcal{W}\subseteq V$ is called a vector subspace of $V$ if
$\forall \alpha ,\beta \in \mathcal{K} ,\:x,y \in \mathcal{W}$ ,

$\alpha *x+\beta *y\in \mathcal{W}$

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Vector space